$ F = \left[\begin{array}{r}4 \\ -2 \\ -1\end{array}\right]$ $ E = \left[\begin{array}{rr}2 & 4 \\ 2 & 2 \\ 0 & -2\end{array}\right]$ Is $ F E$ defined?
Solution: In order for multiplication of two matrices to be defined, the two inner dimensions must be equal. If the two matrices have dimensions $( m \times  n)$ and $( p \times q)$ , then $ n$ (number of columns in the first matrix) must equal $ p$ (number of rows in the second matrix) for their product to be defined. How many columns does the first matrix, $ F$ , have? How many rows does the second matrix, $ E$ , have? Since $ F$ has a different number of columns (1) than $ E$ has rows (3), $ F E$ is not defined.